We investigate geodesic finite elements for functions with values in a space of zero curvature, like a torus or the Möbius strip. Unlike in the general case, a closed-form expression for geodesic finite element functions is then available. This simplifies computations, and allows us to prove optimal estimates for the interpolation error in 1d and 2d. We also show the somewhat surprising result that the discretization by Kirchhoff transformation of the Richards equation proposed in Berninger et al. (SIAM J Numer Anal 49(6):2576–2597, 2011) is a discretization by geodesic finite elements in the manifold with a special metric.